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In mathematics, especially in the field of ring theory, a free ideal ring, or fir, is a ring in which all right ideals are free modules with unique rank. A ring such that all right ideals with at most n generators are free and have unique rank is called an n-fir. A semifir is a ring in which all finitely generated right ideals are free modules of unique rank. The semifir property is left-right symmetric, but the fir property is not.

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